Mean Distance between Two Random Points in Unit Cube

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Theorem

The mean distance $R$ between $2$ points chosen at random from the interior of a unit cube is given by:

\(\ds R\) \(=\) \(\ds \frac {4 + 17 \sqrt 2 - 6 \sqrt3 - 7 \pi} {105} + \frac {\map \ln {1 + \sqrt 2 } } 5 + \frac {2 \, \map \ln {2 + \sqrt 3} } 5\)
\(\ds \) \(\approx\) \(\ds 0 \cdotp 66170 \, 71822 \, 67176 \, 23515 \, 582 \ldots\)


The value $R$ is known as the Robbins constant.


Proof

From Mean Distance between Two Random Points in Cuboid:

Let $B$ be a cuboid in the Cartesian $3$-space $\R^3$ as:

$\size x \le a$, $\size y \le b$, $\size z \le c$

Let $E$ denote the mean distance $D$ between $2$ points chosen at random from the interior of $B$.


Then:

\(\ds E\) \(=\) \(\ds \dfrac {2 r} {15} - \dfrac 7 {45} \paren {\paren {r - r_1} \paren {\dfrac {r_1} a}^2 + \paren {r - r_2} \paren {\dfrac {r_2} b}^2 + \paren {r - r_3} \paren {\dfrac {r_3} c}^2}\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \dfrac 8 {315 a^2 b^2 c^2} \paren {a^7 + b^7 + c^7 - {r_1}^7 - {r_2}^7 - {r_3}^7 + r^7}\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \dfrac 1 {15 a b^2 c^2} \paren {b^6 \sinh^{-1} \dfrac a b + c^6 \sinh^{-1} \dfrac a c - {r_1}^2 \paren { {r_1}^4 - 8 b^2 c^2} \sinh^{-1} \dfrac a {r_1} }\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \dfrac 1 {15 a^2 b c^2} \paren {c^6 \sinh^{-1} \dfrac b c + a^6 \sinh^{-1} \dfrac b a - {r_2}^2 \paren { {r_2}^4 - 8 c^2 a^2} \sinh^{-1} \dfrac b {r_2} }\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \dfrac 1 {15 a^2 b^2 c} \paren {a^6 \sinh^{-1} \dfrac c a + b^6 \sinh^{-1} \dfrac c b - {r_3}^2 \paren { {r_3}^4 - 8 a^2 b^2} \sinh^{-1} \dfrac c {r_3} }\)
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \dfrac 4 {15 a b c} \paren {a^4 \arcsin \dfrac {b c} {r_2 r_3} + b^4 \arcsin \dfrac {a c} {r_3 r_1} + c^4 \arcsin \dfrac {a b} {r_1 r_2} }\)


where:

\(\ds r\) \(=\) \(\ds \sqrt {a^2 + b^2 + c^2}\)
\(\ds r_1\) \(=\) \(\ds \sqrt {b^2 + c^2}\)
\(\ds r_2\) \(=\) \(\ds \sqrt {a^2 + c^2}\)
\(\ds r_3\) \(=\) \(\ds \sqrt {a^2 + b^2}\)


The result follows by setting $a = b = c = \dfrac 1 2$.

Hence we have:

\(\ds r\) \(=\) \(\ds \sqrt {\dfrac 1 {2^2} + \dfrac 1 {2^2} + \dfrac 1 {2^2} }\)
\(\ds \) \(=\) \(\ds \sqrt {\dfrac 3 4}\)
\(\ds \) \(=\) \(\ds \dfrac {\sqrt 3} 2\)

and:

\(\ds r_1 = r_2 = r_3\) \(=\) \(\ds \sqrt {\dfrac 1 {2^2} + \dfrac 1 {2^2} }\)
\(\ds \) \(=\) \(\ds \sqrt {\dfrac 2 4}\)
\(\ds \) \(=\) \(\ds \dfrac {\sqrt 2} 2\)

So:

$r - r_1 = r - r_2 = r - r_3 = \dfrac {\sqrt 3 - \sqrt 2} 2$

Thus:

\(\ds E\) \(=\) \(\ds \dfrac 2 {15} \dfrac {\sqrt 3} 2 - \dfrac 7 {45} \paren {\dfrac {\sqrt 3 - \sqrt 2} 2 \paren {\dfrac {\sqrt 2} {2 a} }^2 + \dfrac {\sqrt 3 - \sqrt 2} 2 \paren {\dfrac {\sqrt 2} {2 b} }^2 + \dfrac {\sqrt 3 - \sqrt 2} 2 \paren {\dfrac {\sqrt 2} {2 c} }^2}\) substituting for all instances of $r$, $r_1$ etc.
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \dfrac 8 {315 a^2 b^2 c^2} \paren {a^7 + b^7 + c^7 - 3 \paren {\dfrac {\sqrt 2} 2}^7 + \paren {\dfrac {\sqrt 3} 2}^7}\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \dfrac 1 {15 a b^2 c^2} \paren {b^6 \sinh^{-1} \dfrac a b + c^6 \sinh^{-1} \dfrac a c - \paren {\dfrac {\sqrt 2} 2}^2 \paren { \paren {\dfrac {\sqrt 2} 2}^4 - 8 b^2 c^2} \sinh^{-1} \dfrac {2 a} {\sqrt 2} }\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \dfrac 1 {15 a^2 b c^2} \paren {c^6 \sinh^{-1} \dfrac b c + a^6 \sinh^{-1} \dfrac b a - \paren {\dfrac {\sqrt 2} 2}^2 \paren { \paren {\dfrac {\sqrt 2} 2}^4 - 8 c^2 a^2} \sinh^{-1} \dfrac {2 b} {\sqrt 2} }\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \dfrac 1 {15 a^2 b^2 c} \paren {a^6 \sinh^{-1} \dfrac c a + b^6 \sinh^{-1} \dfrac c b - \paren {\dfrac {\sqrt 2} 2}^2 \paren { \paren {\dfrac {\sqrt 2} 2}^4 - 8 a^2 b^2} \sinh^{-1} \dfrac {2 c} {\sqrt 2} }\)
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \dfrac 4 {15 a b c} \paren {a^4 \, \map \arcsin {2 b c} + b^4 \, \map \arcsin {2 a c} + c^4 \, \map \arcsin {2 a b} }\)
\(\ds \) \(=\) \(\ds \dfrac 2 {15} \dfrac {\sqrt 3} 2 - \dfrac 7 {45} \paren {\dfrac {\sqrt 3 - \sqrt 2} 2 \paren {\sqrt 2}^2 + \dfrac {\sqrt 3 - \sqrt 2} 2 \paren {\sqrt 2}^2 + \dfrac {\sqrt 3 - \sqrt 2} 2 \paren {\sqrt 2}^2}\) substituting for all instances of $a = b = c = \dfrac 1 2$
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \dfrac 8 {315 \paren {\dfrac 1 2}^2 \paren {\dfrac 1 2}^2 \paren {\dfrac 1 2}^2} \paren {\paren {\dfrac 1 2}^7 + \paren {\dfrac 1 2}^7 + \paren {\dfrac 1 2}^7 - 3 \paren {\dfrac {\sqrt 2} 2}^7 + \paren {\dfrac {\sqrt 3} 2}^7}\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \dfrac 1 {15 \paren {\dfrac 1 2} \paren {\dfrac 1 2}^2 \paren {\dfrac 1 2}^2} \paren {\paren {\dfrac 1 2}^6 \sinh^{-1} 1 + \paren {\dfrac 1 2}^6 \sinh^{-1} 1 - \paren {\dfrac {\sqrt 2} 2}^2 \paren { \paren {\dfrac {\sqrt 2} 2}^4 - 8 \paren {\dfrac 1 2}^2 \paren {\dfrac 1 2}^2} \sinh^{-1} \dfrac 1 {\sqrt 2} }\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \dfrac 1 {15 \paren {\dfrac 1 2}^2 \paren {\dfrac 1 2} \paren {\dfrac 1 2}^2} \paren {\paren {\dfrac 1 2}^6 \sinh^{-1} 1 + \paren {\dfrac 1 2}^6 \sinh^{-1} 1 - \paren {\dfrac {\sqrt 2} 2}^2 \paren { \paren {\dfrac {\sqrt 2} 2}^4 - 8 \paren {\dfrac 1 2}^2 \paren {\dfrac 1 2}^2} \sinh^{-1} \dfrac 1 {\sqrt 2} }\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \dfrac 1 {15 \paren {\dfrac 1 2}^2 \paren {\dfrac 1 2}^2 \paren {\dfrac 1 2} } \paren {\paren {\dfrac 1 2}^6 \sinh^{-1} 1 + \paren {\dfrac 1 2}^6 \sinh^{-1} 1 - \paren {\dfrac {\sqrt 2} 2}^2 \paren { \paren {\dfrac {\sqrt 2} 2}^4 - 8 \paren {\dfrac 1 2}^2 \paren {\dfrac 1 2}^2} \sinh^{-1} \dfrac 1 {\sqrt 2} }\)
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \dfrac 4 {15 \paren {\dfrac 1 2} \paren {\dfrac 1 2} \paren {\dfrac 1 2} } \paren {\paren {\dfrac 1 2}^4 \, \map \arcsin {2 \paren {\dfrac 1 2} \paren {\dfrac 1 2} } + \paren {\dfrac 1 2}^4 \, \map \arcsin {2 \paren {\dfrac 1 2} \paren {\dfrac 1 2} } + \paren {\dfrac 1 2}^4 \, \map \arcsin {2 \paren {\dfrac 1 2} \paren {\dfrac 1 2} } }\)
\(\ds \) \(=\) \(\ds \dfrac {\sqrt 3} {15} - \dfrac {21} {45} \paren {\sqrt 3 - \sqrt 2}\) simplification
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \dfrac {8 \times 2^6} {315} \paren {\dfrac 3 {2^7} - \dfrac {3 \sqrt 2^7} {2^7} + \dfrac {\sqrt 3^7} {2^7} }\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \dfrac {2^5} {15} \paren {\dfrac 1 {2^5} \sinh^{-1} 1 - \dfrac 1 2 \paren {\dfrac 1 4 - \dfrac 1 2} \sinh^{-1} \dfrac 1 {\sqrt 2} }\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \dfrac {2^5} {15} \paren {\dfrac 1 {2^5} \sinh^{-1} 1 - \dfrac 1 2 \paren {\dfrac 1 4 - \dfrac 1 2} \sinh^{-1} \dfrac 1 {\sqrt 2} }\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \dfrac {2^5} {15} \paren {\dfrac 1 {2^5} \sinh^{-1} 1 - \dfrac 1 2 \paren {\dfrac 1 4 - \dfrac 1 2} \sinh^{-1} \dfrac 1 {\sqrt 2} }\)
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \dfrac {4 \times 2^3} {15} \paren {\dfrac 3 {2^4} \arcsin \dfrac 1 2}\)
\(\ds \) \(=\) \(\ds \dfrac {3 \sqrt 3 - 21 \sqrt 3 + 21 \sqrt 2} {45} + \dfrac 4 {315} \paren {3 - 3 \sqrt 2^7 + \sqrt 3^7}\) simplification
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \dfrac 1 5 \sinh^{-1} 1 + \dfrac 4 5 \sinh^{-1} \dfrac 1 {\sqrt 2} - \dfrac 2 5 \arcsin \dfrac 1 2\)
\(\ds \) \(=\) \(\ds \dfrac {21 \sqrt 2 - 18 \sqrt 3} {45} + \dfrac 4 {315} \paren {3 + 27 \sqrt 3 - 24 \sqrt 2}\) simplification
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \dfrac 1 5 \map \ln {1 + \sqrt {1^2 + 1} } + \dfrac 4 5 \map \ln {\dfrac 1 {\sqrt 2} + \sqrt {\paren {\dfrac 1 {\sqrt 2} }^2 + 1} } - \dfrac 2 5 \arcsin \dfrac 1 2\) Inverse Hyperbolic Sine Logarithmic Formulation
\(\ds \) \(=\) \(\ds \dfrac {7 \sqrt 2 - 6 \sqrt 3} {15} + \dfrac 4 {105} \paren {1 + 9 \sqrt 3 - 8 \sqrt 2}\) simplification
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \dfrac 1 5 \map \ln {1 + \sqrt 2} + \dfrac 4 5 \map \ln {\dfrac {1 + \sqrt 3} {\sqrt 2} } - \dfrac 2 5 \dfrac \pi 6\) Sine of $30 \degrees$
\(\ds \) \(=\) \(\ds \dfrac {49 \sqrt 2 - 42 \sqrt 3 + 4 + 36 \sqrt 3 - 32 \sqrt 2 - 7 \pi} {105}\) common denominator
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \dfrac 1 5 \map \ln {1 + \sqrt 2} + \dfrac 2 5 \map \ln {\paren {\dfrac {1 + \sqrt 3} {\sqrt 2} }^2 }\) Logarithm of Power
\(\ds \) \(=\) \(\ds \frac {4 + 17 \sqrt 2 - 6 \sqrt3 - 7 \pi} {105} + \dfrac {\map \ln {1 + \sqrt 2} } 5 + \dfrac {2 \, \map \ln {2 + \sqrt 3} } 5\) tidying up

$\blacksquare$


Sources

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